I checked the unused permits numbers, and if they are any indication, Tuesday and Wednesday start dates appear to be in least demand of any week days. And the two weeks after the July 4th week / weekend are less in demand than the last two weeks of July or the first two of August. Available numbers seem to pick some later in August -- probably due to school starting in mid August for some.
It is likely that Monday through Thursday start days are in less demand than Friday - Sunday.
And you don't need to worry about exit dates or number of days on the trail. Permits ask about planned number of days, but the actual permit limits are only based on the entry date.
But you should put down your first choices on the lottery form, with mid-week or earlier in July as lower choices. If your app gets pulled early, you get the primo slots!
